On the relative Galois module structure of rings of integers in tame extensions

Agboola, Adebisi; McCulloh, Leon

doi:10.2140/ant.2018.12.1823

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On the relative Galois module structure of rings of integers in tame extensions

Adebisi Agboola and Leon R. McCulloh

Vol. 12 (2018), No. 8, 1823–1886

DOI: 10.2140/ant.2018.12.1823

Abstract

Let $F$ be a number field with ring of integers $O_{F}$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl (O_{F} G)$ of $O_{F} G$ that involves applying the work of McCulloh in the context of relative algebraic $K$ theory. For a large class of soluble groups $G$ , including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of $Cl (O_{F} G)$ . This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.

Keywords

Galois module structure, realisable classes, rings of integers, inverse Galois problem, relative K-group.

Mathematical Subject Classification 2010

Primary: 11R33

Secondary: 11R32, 11R65, 19F99

Milestones

Received: 28 August 2015

Revised: 23 March 2018

Accepted: 2 July 2018

Published: 4 December 2018

Authors

Adebisi Agboola
Department of Mathematics University of California, Santa Barbara CA United States

Leon R. McCulloh
Department of Mathematics University of Illinois Urbana, IL United States

Vol. 12, No. 8, 2018